Issue34

F. Berto, Frattura ed Integrità Strutturale, 34 (2015) 11-26; DOI: 10.3221/IGF-ESIS.34.02 13 in an infinitely small element are not possible, they can be approximated with sufficient accuracy by calculating the fracture energy over the entire fractured cross section of an unnotched tensile specimen [35]. Notched components loaded under static loads show that the average ASFE decreases with increasing the notch sharpness, with the ASFE parameter being plotted as a function of the theoretical stress concentration factor, K t , and the temperature [35]. For a common welded structural steel and K t = 1, the ASFE value, obtained by tensile tests, is about 1.0MJ/m 3 while for values of K t greater than 3.0 a plateau value is visible [35]. Depending on the considered welded metal, the plateau approximately ranges between 0.15 and 0.35 MJ/m 3 . These values are not so different from the mean value that characterises the high cycle fatigue strength of welded joints, W c = 0 . 105 MJ/m 3 but with reference to a specific control volume [8, 10]. The criterion based on the energy density factor, S , gave a sound theoretical basis to the experimental findings [32-35] and the approach, used in different fields, was strongly supported by a number of researchers [36]. The concept of strain energy density has also been reported in the literature in order to predict the fatigue behaviour of notches both under uniaxial and multi-axial stresses [37-38]. It should be remembered that in referring to small-scale yielding, a method based on the averaged of the stress and strain product within the elastic-plastic domain around the notch was extended to cyclic loading of notched components [39]. In particular in Ref. [40] it was proposed a fatigue master life curve based on the use of the plastic strain energy per cycle as evaluated from the cyclic hysteresis loop and the positive part of the elastic strain energy density. The two views, cyclic hysteresis loop concept evaluating the plastic energy for tensile specimens [39, 40] and the criterion evaluating the local accumulated SED near the crack tip [28], although formally different, are strictly connected and both tied to the concept of Absorbed Specific Fracture Energy. The averaged strain energy density criterion, proposed in Refs [7-14, 41], states that brittle failure occurs when the mean value of the strain energy density over a control volume (which becomes an area in two dimensional cases) is equal to a critical energy W c . The SED approach is based both on a precise definition of the control volume and the fact that the critical energy does not depend on the notch sharpness. Such a method was formalised and applied first to sharp, zero radius, V-notches and later extended to blunt U- and V-notches under Mode I loading [11] and successfully applied to welded joints [10]. The control radius R 0 of the volume, over which the energy has to be averaged, depends on the ultimate tensile strength, the fracture toughness and Poisson’s ratio in the case of static loads, whereas it depends on the unnotched specimen’s fatigue limit, the threshold stress intensity factor range and the Poisson’s ratio under high cycle fatigue loads. The approach was successfully used under both static and fatigue loading conditions to assess the strength of notched and welded structures subjected to predominant mode I and also to mixed mode loading [7-14]. The extension of the SED approach to ductile fracture is possible, with a major problem being the definition of the control volume and the influence of the dilatational and distortional components of the strain energy density. Recently, the effect of plasticity in terms of strain energy density over a given control volume has been considered by the present authors, showing different behaviours under tension and torsion loading, as well as under small and large scale yielding [14]. Several criteria have been proposed to predict fracture loads of components with notches , subjected to mode I loading [20-21, 42-53]. Recently, fracture loads of notched specimens (sharp and blunted U and V notches) loaded under mode I have been successfully predicted, using a criterion based on the cohesive zone model [54-57], and in parallel by applying the local strain energy density [7-14]. The problem of brittle failure from blunted notches loaded under mixed mode is more complex than in mode I loading and experimental data, particularly for notches with a non- negligible radius, is scarce. The main aim of some recent papers was to generalise the previous results valid for components with blunted notches loaded under mode I, to notched components loaded under mixed mode [58-61]. This generalization is based on the hypothesis that fracture mainly depends on the local mode I and on the maximum value of the principal stress or the strain energy density. The proposal of mode I dominance for cracked plates was suggested first in Ref. [62] when dealing with cracked plates under plane loading and transverse shear, where the crack grows in the direction almost perpendicular to the maximum tangential stress in radial direction from its tip. Two different methods are used to verify such a hypothesis: the cohesive zone model and the model based on the strain energy density over a control volume [58-60]. Both methods allow us to evaluate the critical load under different mixed mode conditions when the material behaviour can be assumed as linear elastic. Dealing with the SED approach it is worth noting that the case of pure compression or combined compression and shear, for example, would require a reformulation for the control radius of the volume, R 0 , and should also take into account the variability of the critical strain energy density W c with respect to the case of uniaxial tension loads. To the best of Author’s knowledge, the first contribution that modifies the total strain energy density criterion (Beltrami’s hypothesys) to account for the different strength properties exhibited by many materials under pure tension and pure compression uniaxial tests was dated 1926 [63]. Dealing with both notched and welded components and summarising the most recent experimental results reported in the literature, the main aim of the present contribution is to present a complete review of the analytical frame of the volume-